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Properties

Label 1-97-97.35-r0-0-0
Degree $1$
Conductor $97$
Sign $-0.574 - 0.818i$
Analytic cond. $0.450466$
Root an. cond. $0.450466$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(97\)
Sign: $-0.574 - 0.818i$
Analytic conductor: \(0.450466\)
Root analytic conductor: \(0.450466\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 97,\ (0:\ ),\ -0.574 - 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1231679098 + 0.2371004704i\)
\(L(\frac12)\) \(\approx\) \(-0.1231679098 + 0.2371004704i\)
\(L(1)\) \(\approx\) \(0.2800573425 + 0.3669659992i\)
\(L(1)\) \(\approx\) \(0.2800573425 + 0.3669659992i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.29798079050760315976916077994, −28.49445932908747895049143055143, −27.73818429050460290666754142858, −26.48194511954583627702875096108, −25.33454203097272926625239484472, −24.04665190744838076627965994609, −23.071402592312867066734453897411, −22.197575036706720655200422220414, −20.41666705592833264356098025337, −19.9675541501170257830541393368, −18.95967679764568308666018021642, −17.57387514574386461957143103018, −17.09784833170861238382056473038, −15.77345495051555874359988455935, −13.53509685368213035482157747586, −12.81625748397168463949935970984, −12.01968956911322912888788583252, −10.7649053395388967449850588353, −9.604555597788181401390234017551, −7.9644341102357127684848655516, −7.34358035107327842946431039613, −5.25373519968485686124174891751, −3.79023790702505352058244326411, −1.90698666573589598036390385095, −0.33517478491711514471991089922, 3.06372744474717554846383685905, 4.787618280889027664224925483203, 5.99617752527181749350427703913, 7.0435646972883074105141072785, 8.663356754215173092133721167946, 9.6738286653837273184901709775, 10.81762467118392484021463028744, 11.89011109817640495872184059619, 13.96569468877486040436588710441, 14.98620638615552273353644642046, 15.958334389862363421027562722829, 16.46251685446609325896385165951, 18.133255494687826306005517933972, 18.681619220813010308616727973286, 20.00687576523278434273690503537, 21.845952962509001722959646001013, 22.35969671986307261270507642080, 23.46835156272115460443222517829, 24.53203007656423246443902207618, 25.96774242831327864482801976502, 26.63899843510630654040090769024, 27.32179813507719363454683073971, 28.57369392791297033553319552519, 29.21164157799950442390013522732, 31.30740881595147322349649766157

Graph of the $Z$-function along the critical line